(1,0,0)-colorability of planar graphs without cycles of length 4 or 6

Abstract

A graph G is (d1, d2, d3)-colorable if the vertex set V (G) can be partitioned into three subsets V1, V2 and V3 such that for i ∈ {1, 2, 3}, the induced graph G[Vi] has maximum vertex-degree at most di. So, (0, 0, 0)-colorability is exactly 3-colorability.The well-known Steinberg's conjecture states that every planar graph without cycles of length 4 or 5 is 3-colorable. As this conjecture being disproved by Cohen-Addad etc. in 2017, a similar question, whether every planar graph without cycles of length 4 or i is 3-colorable for a given i ∈ {6, . . . , 9}, is gaining more and more interest. In this paper, we consider this question for the case i = 6 from the viewpoint of improper colorings. More precisely, we prove that every planar graph without cycles of length 4 or 6 is (1,0,0)-colorable, which improves on earlier results that they are (2,0,0)-colorable and also (1,1,0)-colorable, and on the result that planar graphs without cycles of length from 4 to 6 are (1,0,0)-colorable.